{
  "id": "1712.09318",
  "version": "v1",
  "published": "2017-12-26T18:07:38.000Z",
  "updated": "2017-12-26T18:07:38.000Z",
  "title": "Formulae for the conjugate and the $\\varepsilon$-subdifferential of the supremum function",
  "authors": [
    "Pedro Pérez-Aros"
  ],
  "comment": "submitted",
  "categories": [
    "math.OC"
  ],
  "abstract": "The aim of this work is to provide formulae for the $\\varepsilon$-subdifferential and the conjungate function of the supremun function $f=\\sup_{t\\in T} f_t$, where $T$ is an index set. The work is principally motivated by the case when the functions $\\{f_t\\}_{t\\in T}$ are lower semicontinuous proper and convex functions. Nevertheless, we explore the case when the family of functions is arbitrary, but they satisfy the relations $f^{\\ast \\ast}=\\sup_{t \\in T} f^{\\ast \\ast}_t$, where $(\\cdot)^{\\ast \\ast}$ denotes the biconjugate of a function. The study focuses its attention on when the space $X$ is finite-dimensional, in this case the formulae can be simplified under certain qualification conditions. However, we show how to extend these finite-dimesional results to arbitrary locally convex spaces without any qualification condition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-26T18:07:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "90C25",
      "90C34",
      "46N10"
    ],
    "keywords": [
      "supremum function",
      "subdifferential",
      "qualification condition",
      "arbitrary locally convex spaces",
      "conjungate function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}